A square loop of side \(2.0\,cm\) is placed inside a long solenoid that has \(50\) turns per centimetre and carries a sinusoidally varying current of amplitude \(2.5\,A\) and angular frequency \(700\,rad\,s ^{-1}\). The central axes of the loop and solenoid coincide. The amplitude of the emf induced in the loop is \(x \times 10^{-4} V\). The value of \(x\) is \(.........\)(Take, \(\pi=\frac{22}{7}\))

The electric fields of two plane electromagnetic plane waves in vacuum are given by \(\overrightarrow{\mathrm{E}}_{1}=\mathrm{E}_{0} \hat{\mathrm{j}} \cos (\omega \mathrm{t}-\mathrm{kx})\) and \(\overrightarrow{\mathrm{E}}_{2}=\mathrm{E}_{0} \hat{\mathrm{k}} \cos (\omega \mathrm{t}-\mathrm{ky})\) At \(t=0,\) a particle of charge \(q\) is at origin with a velocity \(\overrightarrow{\mathrm{v}}=0.8 \mathrm{c} \hat{\mathrm{j}}\) (\(c\) is the speed of light in vacuum). The instantaneous force experienced by the particle is 

The length of a potentiometer wire is \(1200\; \mathrm{cm}\) and it carries a current of \(60 \;\mathrm{mA}\). For a cell of \(emf\;5\; \mathrm{V}\) and intemal resistance of \(20\; \Omega,\) the null point on it is found to be a \(1000\; \mathrm{cm} .\) The resistance of whole wire is .............. \(\Omega\)

The following observations were taken for determining surface tension \(T\) of water by capillary method: diameter of capillary, \(D= 1.25 \times 10^{-2}\; m\) rise of water, \(h=1.45 \times 10^{-2}\; m \)  Using \(g= 9.80 \;m/s^2\) and the simplified relation \(T = \frac{{rhg}}{2}\times 10^3 N/m\) , the possible error in surface tension is ........... \(\%\)

The position of a particle as a function of time \(t\), is given by \(x\left( t \right) = at+ b{t^2} - c{t^3}\) where \(a, b\) and \(c\) are constants. When the particle attains zero acceleration, then its velocity will be

Match the thermodynamic processes taking place in a system with the correct conditions. In the table: \(\Delta Q\) is the heat supplied, \(\Delta W\) is the work done and \(\Delta U\) is change in internal energy of the system

Process	Condition
\((I)\) Adiabatic	\((A)\; \Delta W =0\)
\((II)\) Isothermal	\((B)\; \Delta Q=0\)
\((III)\) Isochoric	\((C)\; \Delta U \neq 0, \Delta W \neq 0 \Delta Q \neq 0\)
\((IV)\) Isobaric	\((D)\; \Delta U =0\)

The explosive in a Hydrogen bomb is a mixture of \({ }_1 \mathrm{H}^2,{ }_1 \mathrm{H}^3\) and \({ }_3 \mathrm{Li}^6\) in some condensed form. The chain reaction is given by \( { }_3 \mathrm{Li}^6+{ }_0 \mathrm{n}^1 \rightarrow{ }_2 \mathrm{He}^4+{ }_1 \mathrm{H}^3 \) \( { }_1 \mathrm{H}^2+{ }_1 \mathrm{H}^3 \rightarrow{ }_2 \mathrm{He}^4+{ }_0 \mathrm{n}^1\) During the explosion the energy released is approximately [Given : \(\mathrm{M}(\mathrm{Li})=6.01690\  \mathrm{amu} . \mathrm{M}\left({ }_1 \mathrm{H}^2\right)=2.01471 \  amu.\) \(\mathrm{M}\left({ }_2 \mathrm{He}^4\right)=4.00388\  \mathrm{amu}\),\( and\ \)\(1 \ \mathrm{amu}=931.5\) \(\mathrm{MeV}]\)

Let \(f : R \to R\) be defined by \(f\left( x \right) = \frac{{\left| x \right| - 1}}{{\left| x \right| + 1}}\) then \(f\) is

In the given circuit, the terminal potential difference of the cell is _______.

If \(\frac{3+i \sin \theta}{4-i \cos \theta}, \theta \in[0,2 \pi],\) is a real number, then an argument of \(\sin \theta+\mathrm{i} \cos \theta\) is

A block of mass \(10\, kg\) starts sliding on a surface with an initial velocity of \(9.8\, ms ^{-1}\). The coefficient of friction between the surface and bock is \(0.5\). The distance covered by the block before coming to rest is: [use \(g =9.8\, ms ^{-2}\) ].........\(m\)

The common tangent to the circles \(x^2 + y^2 = 4\) and \(x^2 + y^2 + 6x + 8y - 24 = 0\) also passes through the point

\(\alpha=\sin 36^{\circ}\) is a root of which of the following equation